Problem: Rui is a professional deep water free diver. His altitude (in meters relative to sea level), $x$ seconds after diving, is modeled by: $d(x)=\dfrac12x^2 -10x$ How many seconds after diving will Rui reach his lowest altitude?
Explanation: Rui's altitude is modeled by a quadratic function, whose graph is a parabola. The lowest altitude is reached at the vertex. So in order to find when that happens, we need to find the vertex's $x$ -coordinate. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} d(x)&=0 \\\\ \dfrac12x^2 -10x&=0 \\\\ x^2-20x&=0 \\\\ x(x-20)&=0 \\\\ \swarrow &\searrow \\\\ x=0\text{ or }&x-20=0 \\\\ x={0}\text{ or }&x={20} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({0})+({20})}{2}=\dfrac{20}{2}=10$ In conclusion, Rui will reach his lowest altitude $10$ seconds after diving.